Lorentz for space & time both relative?

Two Lorentz transformations based on relative space with absolute time and absolute space with relative time were presented here. Now we look at beginning with space and time both relative, in two different ways.

R-R Case 1

This case begins with: r´ = r – vt and t´ = t – r/v.

Consider then a linear function of these:

r´ = γ(r – vt) and t´ = γ(t – r/v)

along with the standard (characteristic) speed c such that r = ct and r´ = ct´. Combine this with r´ to get